Complex numbers, conjugate, absolute value, Argand diagram, Euler relation, De Moivre theorem, powers, roots, factorization of a polynomial. Vector spaces, subspaces, sum of subspaces, subspace generated by a set of vectors, linear independence, basis, dimension. Matrices, operations, inverse, transpose, composite matrices, row space, rank, row echelon form, triangular, symmetric, hermitian, orthogonal matrices, trace, similar matrices, row equivalence, change of basis, linear systems. Determinants, properties, Laplace expansion formula, determinant of a triangular matrix, adjoint-inverse, Cramer’s rule. Characteristic polynomial, Cayley-Hamilton theorem, eigenvalues-eigenvectors (properties for symmetric, orthogonal matrices), functions of matrices. Linear mappings, kernel, image, matrix associated with a linear map, rotations, change of basis of a linear map. Diagonalization of a matrix, functions of diagonalizable matrices, diagonalization of a hermitian matrix, quadratic forms. Second order linear differential equations.

After the successful fulfilment of the course, the student:

• will have a deep and working knowledge of the theory of linear spaces, the theory of matrices and determinants,
• will have the knowledge of more advanced and important issues of Linear Algebra, such as the theory of eigenvalues-eigenvectors, of linear mappings and diagonalization,
• will have the ability to treat the notions of linearly dependent and independent vectors, of the basis and dimension of a linear space of subspace,
• will have the ability to perform calculations with matrices, to use the technique of row-equivalence for various purposes and to solve linear systems of equations,
• will have the ability to compute determinants with various methods and in various dimensions through recursion relations,
• will have the skills to represent a linear mapping with its matrix and compute various quantities, as well as to perform its diagonalization,
• will have the ability to solve simple differential equations of second order.

Not required.

1. Linear Algebra, Theory and Applications, G. Donatos and M. Adam, Gutenberg Eds. 2. Linear Algebra, S. Lang, Springer.

Linear Algebra, S. Andreadakis, Symmetria Eds.

Systematic development and explanation of the theory (and through examples), methods of solutions of exercises, solutions of exercises in the teaching hours and in the problem session hours, final written exam.

Homeworks, tests, final written exam.

Face-to-face